ARMA system identification via the Cholesky least squares method

Abstract

A computationally efficient method for the identification of scalar autoregressive moving average (ARMA) models of the form a(z)y = b(z)u+ c(z)\epsilon is introduced; it is called the Cholesky Least Squares (CLS) method. This technique iteratively estimates the coefficients of the polynomials a(z) and b(z) by using the least squares method on data which, at each iteration step, are a filtered version of the original observations \{y_{1},...,y_{N}; u_{1},...,u_{N}\} . The filter employed at each stage is the inverse of the current estimate of c(z) and this estimate is generated by factoring the sample covariance matrix of the residual sequence by using a fast Cholesky factorization algorithm [6], [7]. We describe a natural extension of the CLS method for the identification of multivariable ARMA systems and present computational experiments demonstrating the operation of this extended version of the algorithm. Our method is a variant of the Generalized Least Squares method [1]-[3], and computational experiments comparing a particular version of this method with the CLS algorithm are presented. Finally, some evidence is presented to support the view that the CLS algorithm, like many other identification methods, computes approximations to the true system's impulse response when it is provided with a (possibly incorrect) set of orders for the polynomials a(z), b(z), c(z) .